On interval and cyclic interval edge colorings of
(3,5)-biregular graphs
Carl Johan Casselgren, Petros A. Petrosyan and Bjarne Toft
The self-archived postprint version of this journal article is available at Linköping University Institutional Repository (DiVA):
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-141695
N.B.: When citing this work, cite the original publication.
Casselgren, C. J., Petrosyan, P. A., Toft, B., (2017), On interval and cyclic interval edge colorings of (3,5)-biregular graphs, Discrete Mathematics, 340(11), 2678-2687.
https://doi.org/10.1016/j.disc.2016.09.020
Original publication available at:
https://doi.org/10.1016/j.disc.2016.09.020
Copyright: Elsevier
On interval and cyclic interval edge colorings of
(3, 5)-biregular graphs
Carl Johan Casselgren
∗Department of Mathematics
Link¨oping University
SE-581 83 Link¨oping, Sweden
Petros A. Petrosyan
†Department of Informatics
and Applied Mathematics
Yerevan State University
0025, Armenia
Bjarne Toft
‡Department of Mathematics
University of Southern Denmark
DK-5230 Odense, Denmark
Dedicated to Horst Sachs on his 90th birthday.Abstract. A proper edge coloring f of a graph G with colors 1, 2, 3, . . . , t is called an interval coloring if the colors on the edges incident to every vertex of G form an interval of integers. The coloring f is cyclic interval if for every vertex v of G, the colors on the edges incident to v either form an interval or the set {1, . . . , t} \ {f (e) : e is incident to v} is an interval. A bipartite graph G is (a, b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b; it has been conjectured that all such graphs have interval colorings. We prove that every (3, 5)-biregular graph has a cyclic interval coloring and we give several sufficient conditions for a (3, 5)-biregular graph to admit an interval coloring.
1
Introduction
An interval coloring (or consecutive coloring) of a graph G is a proper coloring of the edges by positive integers such that the colors on the edges incident to any vertex of G form an interval of integers. The notion of interval colorings was introduced by Asratian and Kamalian [6] (available in English as [5]), motivated by the problem of finding compact school timetables, that is, timetables such that the lectures of each teacher and each class are scheduled at
∗E-mail address: carl.johan.casselgren@liu.se Part of the work done while the author was a postdoc at
University of Southern Denmark. Research supported by SVeFUM.
†E-mail address: pet petros@ipia.sci.am ‡E-mail address: btoft@imada.sdu.dk
consecutive periods. Hansen [14] described another scenario (originally suggested by Jesper Bang-Jensen). A school wishes to schedule parent-teacher conferences in time slots so that every person’s conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for the people and edges for the required meetings has an interval coloring.
All regular bipartite graphs have interval colorings, since they decompose into perfect matchings. Not every graph has an interval coloring, since a graph G with an interval coloring must have a proper ∆(G)-edge-coloring [6], where ∆(G) denotes the maximum degree of a graph G. Sevastjanov [26] proved that determining whether a bipartite graph has an interval coloring is N P-complete. Nevertheless, trees [18, 14, 5], complete bipartite graphs [18, 14, 6], grids [13], and outerplanar bipartite graphs [12, 7] all have interval colorings. Giaro [11] showed that one can decide in polynomial time whether bipartite graphs with maximum degree 4 have interval 4-colorings. The smallest known maximum degree of a bipartite graph without an interval coloring is 11 [23].
A bipartite graph with parts X, Y is called (a, b)-biregular if all vertices of X have degree a and all vertices of Y have degree b. In this paper we study the following well-known conjecture [17, 27] for the case (a, b) = (3, 5):
Conjecture 1.1. Every (a, b)-biregular graph has an interval coloring.
By results of [14] and [16], all (2, b)-biregular graphs admit interval colorings (the result for odd b was obtained independently by Kostochka [19]). Hanson and Loten [15] proved that no (a, b)-biregular graph has an interval coloring with fewer than a + b − gcd(a, b) colors, where gcd denotes the greatest common divisor.
Several sufficient conditions for a (3, 4)-biregular graph G to admit an interval 6-coloring have been obtained: Pyatkin [25] proved that if G has a 3-regular subgraph covering the vertices of degree 4, then it has an interval coloring; Yang et al. [28] proved that if G is the union of two edge-disjoint (2, 3)-biregular subgraphs H1 and H2 such that vertices of degree
4 in G has degree 2 in H1 and H2, then G has an interval coloring; G has an interval coloring
if it has a spanning subgraph consisting of paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8} [4, 10]. (See also [3, 8] for related results.) However, it is still an open question whether every (3, 4)-biregular graph has an interval coloring. In [9] the first and third author proved that every (3, 6)-biregular graph has an interval 7-coloring; by the result in [2] the number of colors is best possible.
It is unknown whether all (3, 5)-biregular graphs have interval colorings; to the best of our knowledge no non-trivial condition implying interval colorings of such graphs is known. By the result of Hanson and Loten [15] we need at least 7 colors for an interval coloring of a (3, 5)-biregular graph. In this paper we present a technique for constructing interval 7-colorings of families of (3, 5)-biregular graphs using so-called MP -subgraphs (defined in Section 3 below). Using this technique we give several sufficient conditions for a (3, 5)-biregular graph to admit an interval 7-coloring. Moreover, we present infinite families of (3, 5)-biregular graphs satisfying our conditions.
First, we consider cyclic interval colorings. A proper edge coloring f of a graph G with colors 1, 2, 3, . . . , t is cyclic interval if for every vertex v of G {f (e) : e is incident to v} or {1, . . . , t} \ {f (e) : e is incident to v} is an interval. Cyclic interval colorings are studied in e.g. [21, 20, 24]. In particular, the general question of determining whether a bipartite
graph G has a cyclic interval coloring is N P-complete [20] and there are concrete examples of connected bipartite graphs having no cyclic interval coloring [21]. Trivially, any bipartite graph with an interval coloring also has a cyclic interval coloring with ∆(G) colors, but the converse does not hold [21]. This means, in particular, that Conjecture 1.1 has the following weaker consequence for which the answer is unknown.
Conjecture 1.2. Every (a, b)-biregular graph has a cyclic interval max{a, b}-coloring.
Note that all bipartite graphs G with ∆(G) − δ(G) ≤ 1 admit cyclic interval colorings [9], where δ(G) as usual denotes the minimum degree in G; so the smallest unsolved case of Conjecture 1.2 is (a, b) = (3, 5). In [9] the first and third author proved that all (4, 8)-biregular graphs have cyclic interval 8-colorings. In this paper we prove that all (3, 5)-biregular graphs have cyclic interval 6-colorings, and we give several sufficient conditions for a (3, 5)-biregular graph to admit such a coloring with 5 colors.
The rest of the paper is organized as follows. In Section 2 we consider cyclic interval colorings, Section 3 contains sufficient conditions for interval coloring (3, 5)-biregular graphs and in Section 4 we discuss these conditions and provide infinite families of graphs satisfying our conditions.
2
Cyclic interval edge colorings
In this section we prove our results on cyclic interval colorings. First we introduce some notation and also state some preliminary results. Throughout the paper, we use the notation G = (X, Y ; E) for a bipartite graph G with bipartition (X, Y ) and edge set E = E(G). We use the convention that if G = (X, Y ; E) is (a, b)-biregular, then the vertices in X have degree a. We denote by dG(v) the degree of a vertex v in G, and by NG(v) the set of vertices
adjacent to v in G. If V′ ⊆ V (G), then N
G(V′) =Sv∈V′NG(v).
For an edge coloring ϕ of a graph G, let M(ϕ, i) = {e ∈ E(G) : ϕ(e) = i}. We define Gϕ(a, b) = G[M(ϕ, a) ∪ M(ϕ, b)]. If e ∈ M(ϕ, i), then e is colored i under ϕ. If ϕ is a
proper t-edge coloring of G and 1 ≤ a, b ≤ t, then a path (cycle) in Gϕ(a, b) is called a
ϕ − (a, b)-colored path (cycle) in G. We also say that such a path or cycle is ϕ-bicolored. By switching colors a and b on a connected component of Gϕ(a, b), we obtain another proper
t-edge coloring of G. We call this operation a ϕ-interchange. For a vertex v ∈ V (G), we say that a color i appears at v under ϕ if there is an edge e incident to v with ϕ(e) = i, and we set
ϕ(v) = {ϕ(e) : e ∈ E(G) and e is incident to v}.
If c /∈ ϕ(v), then c is missing at v under ϕ. Moreover, if ϕ(v) = {c}, that is, ϕ(v) is singleton, then ϕ(v) usually denotes the color c rather than the set {c}. In all the above definitions, we often leave out the explicit reference to a coloring ϕ, if the coloring is clear from the context. We shall say that a proper t-edge coloring ϕ of a graph G using positive integers as colors is interval at a vertex v ∈ V (G) if the colors on the edges incident to v form a interval of integers. The coloring ϕ is cyclic interval at a vertex v ∈ V (G), if it is interval at v or if the set {1, . . . , t} \ {ϕ(e) : e is incident to v} is an interval of integers.
A mixed graph is a graph containing both directed and undirected edges. We denote a mixed graph G by G = (V ; E, A), where E = E(G) are the (undirected) edges of G and
A = A(G) are the directed edges or arcs of G. For a mixed graph G = (V ; E, A), and a subset E′ ⊆ E an orientation of E′
is the mixed graph obtained from G by orienting each edge of E′
.
Let us now prove the main result of this section.
Theorem 2.1. Every (3, 5)-biregular graph has a cyclic interval 6-coloring.
Proof. Let G = (X, Y ; E) be a (3, 5)-biregular graph. Since G is (3, 5)-biregular, |X| = 5k
and |Y | = 3k for some positive integer k. From G, we form a new graph H by adding a set X′
of k new vertices, where each such vertex is adjacent to exactly three vertices in Y , and each vertex in Y is adjacent to exactly one vertex of X′
. The resulting graph H is (3, 6)-biregular and thus has an interval 7-coloring f [9]. By taking all colors modulo 6 we obtain a cyclic interval 6-coloring of H; the restriction of this coloring to G is a cyclic interval 6-coloring of G.
In the remaining part of this section we consider cyclic interval 5-colorings. To this end, let us define a particular kind of subgraph of a (3, 5)-biregular graph: an MF -subgraph of a (3, 5)-biregular graph G = (X, Y ; E) is a pair (M, F ) consisting of a matching M covering Y , and a subgraph F of G with maximum degree 2, edge-disjoint from M, and such that dF(v) = 2 if and only if
(a) v ∈ Y , or
(b) v ∈ X and v is not covered by M.
The following proposition characterizes (3, 5)-biregular graphs having a cyclic interval 5-coloring.
Theorem 2.2. A (3, 5)-biregular graph has a cyclic interval 5-coloring if and only if it has an MF -subgraph.
Proof. Let G = (X, Y ; E) be a (3, 5)-biregular graph. Necessity is straightforward: let ϕ be
a cyclic interval 5-coloring of G. We define M = M(ϕ, 4) and F = G[M(ϕ, 1) ∪ M(ϕ, 2)]. Then (M, F ) is an MF -subgraph of G.
For sufficiency, suppose that (M, F ) is an MF -subgraph of G. We shall construct a cyclic interval 5-coloring g of the graph G in several steps. We set H = G − E(F ) and for every edge e ∈ M, we set g(e) = 4.
In the following we shall define a proper edge coloring ϕ of H′
= H − M using colors 3, 5 such that for each component P = x1y1x2. . . xk in F that is a path, if 3 appears at x1 under
ϕ, then 5 appears at xk under ϕ. Let XF be the set of all endpoints of components in F that
are paths. Let XP ⊆ X be the set of all vertices x ∈ X \ XF such that x is an endpoint of a
component Q in H′ that is a path with one endpoint in X F.
We now define the graph T by setting V (T ) = XF ∪ XP and letting xx′ ∈ E(T ) if and
only if
• x and x′
are distinct endpoints of a component in F , or • x and x′
are distinct endpoints of a component in H′
Note that for each component of H′
that is a path, either both or none of its endpoints are in T .
Claim. T is bipartite.
Proof. Each vertex in XF is an endpoint of exactly one path in F and also an endpoint of
exactly one path in H′
= H − M. Each vertex in XP has degree one in T . Hence, each
component in T is a cycle of even length or a path.
Let c : V (T ) → {A, B} be a proper vertex coloring of T . We now define the proper edge coloring ϕ of H′
by alternately using colors 3 and 5 on each component Q of H′
; if Q is path with endpoints in T , then we begin by color 3 on the edge incident to a vertex which is colored A under c; if Q is a path with no endpoint in T or a cycle, then we color its edges properly with 3 and 5 in any of the two possible ways.
Note that for each component P = x1y1x2. . . xk in F that is a path, if 3 appears at x1
under ϕ, then 5 appears at xk under ϕ. We set g(e) = ϕ(e), for any edge e ∈ E(H′).
It remains to color the edges of F . For any cycle C of F , color its edges properly using colors 1, 2. Since 3 or 5 appears at each vertex of C, we get a coloring that is cyclic interval at each vertex of C.
Suppose now that P = x1y1x2. . . xkis a component of F that is a path and that 3 appears
at x1 under g. Then 5 appears at xk. We extend g to the edges of P by properly coloring
its edges alternately using colors 1 and 2, and starting with color 2 at x1. Since 4 appears
at the endpoints of P and 3 or 5 appears at x2, . . . , xk−1, this yields a coloring that is cyclic
interval at every vertex of P .
We conjecture that every (3, 5)-biregular graph has an MF -subgraph, and thus also a cyclic interval 5-coloring. Our next result establishes existence of MF -subgraphs for two classes of (3, 5)-biregular graphs.
Theorem 2.3. Let G be a (3, 5)-biregular graph such that either
(i) G has a 3-regular subgraph covering all vertices of degree 5, or (ii) G has a (3, 4)-biregular subgraph covering all vertices of degree 5.
Then G has a cyclic interval 5-coloring.
For the proof of Theorem 2.3, we shall use the following theorem of Ore [22] (see also Theorem 7.2.2 in [1]). For a graph G and a function f : V (G) → {0, 1, 2, . . . }, an f -factor of G is a spanning subgraph F of G, such that f (v) = dF(v) for each vertex v ∈ V (G). For a
subset U ⊆ V (G), we denote by EG(U, v) the set of edges whose ends are v and some vertex
from U. As usual, we define f (U) =P
v∈Uf (v) when U ⊆ V (G).
Theorem 2.4. [22] Let G be a bipartite graph with parts X and Y . Then G has an f -factor
if and only if f (X) = f (Y ) and for any set U ⊆ X
f (U) ≤X
y∈Y
Proof of Theorem 2.3. Let G = (X, Y ; E) be a (3, 5)-biregular graph and suppose that (i) of
Theorem 2.3 holds. Then there is a 3-regular subgraph L ⊆ G that covers the vertices of degree 5 in G. By Hall’s condition, there is a perfect matching M in L. Put L′
= L − M. Now, consider the graph J = G − E(L). Since any vertex of degree 5 in G has degree 2 in J, there is by Hall’s theorem a matching MJ in J covering all vertices of degree 3 in G. Let
Y′
be the set of all vertices in Y that are covered by MJ. Next, let M′ be a matching in L′
such that M′
covers Y′
and no vertex in Y \ Y′
is covered by M′
. Let F be the subgraph of G induced by M′∪ E(J) \ M
J. Then any vertex in Y has degree 2 in F , and a vertex x ∈ X
has degree 2 in F if and only if x is not covered by M. Thus (M, F ) is an MF -subgraph of G, so G has a cyclic interval 5-coloring by Theorem 2.2.
Suppose now that (ii) holds. Let L be a (3, 4)-biregular subgraph of G such that all vertices of degree 5 in G has degree 4 in L. Obviously L is obtained from G by removing a set X′ ⊆ X of vertices such that
• NG(X′) = Y , and
• if x, x′ ∈ X′
and x 6= x′
, then NG(x) ∩ NG(x′) = ∅.
We define a function f on V (G) by setting f (u) = ( 1, if u ∈ X \ X′ , 2, if u ∈ X′ or u ∈ Y .
Since |X| = 5k, |X′| = k and |Y | = 3k, Theorem 2.4 for the graph G implies that there is
an f -factor of G, i.e., a subgraph K such that (a) dK(y) = 2 for each vertex y ∈ Y ;
(b) dK(x) = 2 for each vertex x ∈ X′;
(c) dK(x) = 1 for each vertex x ∈ X \ X′.
We shall now construct an MF -subgraph of G from K. Since vertices in X′
have disjoint neighborhoods, the edges in E(G) \ E(K) incident to vertices in X′
form a matching M′
. Denote by ˆY the vertices in Y that are not covered by M′
. Since each vertex x ∈ X′
has degree 2 in K, no vertex in ˆY is adjacent to a vertex in X′
in G − E(K) − M′
. Thus each vertex in ˆY has degree 3 in G − E(K) − M′
, so by Hall’s condition, there is a matching ˆM covering ˆY in G − E(K) − M′
. By setting M = M′∪ ˆ
M we obtain a matching that covers Y . Now, define the graph F by letting it be the subgraph of G induced by E(G) \ (E(K) ∪ M). Each vertex in Y has degree 2 in F , and a vertex x ∈ X has degree 2 in F if and only if x is not covered by M. Thus (M, F ) is an MF -subgraph of G, so G has a cyclic interval 5-coloring by Theorem 2.2.
3
Interval edge colorings from
M P -subgraphs
In this section we consider interval coloring of (3, 5)-biregular graphs. We shall use the fol-lowing lemma establishing a relation between proper 5-edge colorings and interval 7-colorings of (3, 5)-biregular graphs.
Lemma 3.1. Let G = (X, Y ; E) be a (3, 5)-biregular graph. Then G has an interval 7-coloring if and only if there is a proper 5-edge 7-coloring f of G such that
(i) for each vertex x ∈ X
f (x) ∈ {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {1, 4, 5}, {1, 2, 5}},
(ii) there are no three vertices x, x′ ∈ X and y ∈ Y forming a 3-vertex-path, such that
f (x) = {1, 2, 3}, f (x′
) = {1, 2, 5}, f (xy) = 1 and f (x′
y) = 2.
Proof. If G has an interval 7-coloring, then we get the required 5-edge coloring f by taking
all colors modulo 5. Conversely, if G has a proper 5-edge coloring satisfying (i) and (ii), then for each vertex x ∈ X such that f (x) ∈ {{1, 4, 5}, {1, 2, 5}}, we recolor the edges incident with x colored 1 and 2 by colors 6 and 7, respectively. This yields an edge coloring that is interval at every vertex of X. Since (ii) holds, this coloring is also interval at every vertex of Y .
Let us now define the notion of an MP -subgraph of a (3, 5)-biregular graph.
Definition 3.2. Let G be a (3, 5)-biregular graph with parts X and Y , where vertices in X have degree 3. An MP -subgraph of G is a pair (M, P ) consisting of a matching M covering Y , and a subgraph P of G, edge-disjoint from M, whose components are paths of length 6 with endpoints in X, such that x ∈ X is not covered by M if and only if x has degree 2 in P . An MP -subgraph (M, P ) is proper if for any component Q of G − M ∪ E(P ) which is a path, it holds that if one endpoint of Q is covered by M, then the other endpoint of Q is also covered by M.
Note that since G is (3, 5)-biregular, |X| = 5k and |Y | = 3k for some positive integer k. So M covers exactly 3k vertices in X, which means that 2k vertices in X have degree 2 in P . This implies that each vertex in Y has degree 2 in P ; and, additionally, the graph G − M ∪ E(P ) has maximum degree 2. Note further that any MP -subgraph of G is an MF -subgraph, but the converse does not hold.
We shall describe a method for constructing interval 7-colorings from MP -subgraphs of (3, 5)-biregular graphs satisfying various conditions; in particular, we establish that every (3, 5)-biregular graph having a proper MP -subgraph admits an interval 7-coloring. In Section 4 we present infinitely many (3, 5)-biregular graphs having proper MP -subgraphs. We also give a sufficient condition for the existence of a proper MP -subgraph of a (3, 5)-biregular graph.
Definition 3.3. For an MP -subgraph (M, P ) of a (3, 5)-biregular graph G = (X, Y ; E), the P -graph, denoted GP, is the graph with vertex set X ∩ V (P ), where x and x′ are adjacent if
and only if
(i) x and x′ are distinct endpoints of one component in G − M ∪ E(P ) which is a path, or
(ii) x and x′ are distinct internal vertices of one component of P , or
(iii) x and x′
In the above definition multiple edges are allowed, i.e. two vertices of GP may be joined
by two parallel edges. Edges of GP are either of type (i), (ii) or (iii), and will be referred to
as green, blue and red edges, respectively.
Let G = (X, Y ; E) be a (3, 5)-biregular graph and (M, P ) an MP -subgraph of G. Given a proper edge coloring ϕ of P using colors 1 and 2, the oriented P-graph, denoted ~GP(ϕ), is
the mixed graph obtained from GP by retaining the color of each edge, and orienting each
red or blue edge xx′ ∈ E(G
P) towards x′ if and only if the first edge of the (1, 2)-colored
path in P with origin x and terminus x′
is colored 1.
Since the graph G − M ∪ E(P ) has maximum degree 2, each vertex in GP is adjacent to
exactly two edges, where one edge is of type (i), and the other one is of type (ii) or (iii), so GP is a union of disjoint even cycles (including 2-cycles). If ϕ is a proper edge coloring of
P with colors 1 and 2, then in ~GP(ϕ), each vertex is incident to exactly one directed edge
and one undirected edge. Moreover, since ~GP(ϕ) is uniquely defined by GP and a proper
coloring ϕ of P with colors 1 and 2, any orientation D of the red edges in GP induces a proper
coloring ϕ of P with colors 1 and 2, and thus also an orientation of the blue edges in D, from which we obtain the graph ~GP(ϕ). A ϕ-interchange on a component T of P corresponds to
changing the direction of one red and one blue edge with ends in T . A vertex coloring ψ of ~GP(ϕ) with colors 3 and 5 is correct if
(a) each green edge has exactly one end colored 3 (and one end colored 5); (b) for each red arc (x, x′
) of ~GP(ϕ), ψ(x) = 5 and ψ(x′) = 3;
(c) there is no blue arc (x, x′) such that ψ(x) = 3 and ψ(x′) = 5.
A sequence Q = v0e1v1. . . vk−1ekvk in a mixed graph ~GP(ϕ) is an undirected path (or just
path) if eiis a directed or undirected edge connecting the vertices vi−1 and vi, i = 1, . . . , k, and
the vertices v0, . . . , vk are distinct. Note that an undirected path in a mixed graph without
multiple edges is uniquely determined by a sequence of vertices. Thus we will often refer to paths as sequences of vertices, when there is no risk of confusion.
Lemma 3.4. Let G be a (3, 5)-biregular graph with an MP -subgraph (M, P ). Then G has an interval 7-coloring if there is a proper coloring ϕ with colors 1, 2 of P such that ~GP(ϕ)
has a correct vertex coloring.
Proof. Let G = (X, Y ; E) be a (3, 5)-biregular graph with an MP -subgraph (M, P ). Let
ϕ be a proper coloring of P with colors 1 and 2, and ψ a correct coloring of ~GP(ϕ). As
noted above, we have that |X| = 5k and |Y | = 3k for some positive integer k, and that M covers exactly 3k vertices in X; consequently, there are 2k vertices in X that have degree 2 in P . Since |Y | = 3k, this means that each vertex in Y has degree 2 in P ; so the graph G − M ∪ E(P ) has maximum degree 2.
Note further that each maximal path in G − M ∪ E(P ) corresponds to a green edge in ~
GP(ϕ). Since ψ is correct, each such a maximal path has one endpoint colored 3 under ψ
and one endpoint colored 5 under ψ.
We define a proper edge coloring g of G − E(P ) by coloring each edge of M with color 4 and properly coloring the edges of G − M ∪ E(P ) with colors 3 and 5 so that for each
component T in G − M ∪ E(P ) that is a path, we begin by color 3 at an endpoint of T colored 3 under ψ.
Let f be the proper 5-edge coloring of G induced by g and ϕ. We shall prove that f satisfies condition (i) and (ii) of Lemma 3.1 thus proving that there is an interval 7-coloring of G. If a vertex x ∈ X is not in P , then f (x) = {3, 4, 5}. If x ∈ X is an endpoint of a component in P , then g(x) = {4, 5} if ϕ(x) = 1, and g(x) = {3, 4} if ϕ(x) = 2, because ψ is correct. If x ∈ X is an internal vertex of a component in P , then ϕ(x) = {1, 2} and g(x) = 3 or g(x) = 5. Thus we conclude that (i) holds.
Consider now condition (ii). Suppose that there are vertices x, x′
, y such that f (x) = {1, 2, 3}, f (x′
) = {1, 2, 5}, f (xy) = 1 and f (x′
y) = 2. Then x and x′
are internal vertices of some component T of P . Furthermore, (x, x′
) is a blue arc in ~GP(ϕ) with ψ(x) = 3 and
ψ(x′
) = 5, which contradicts that ψ is correct. Thus (ii) holds and consequently there is an interval 7-coloring of G.
A sequence Q = v0e1v1. . . vk−3ek−2vk−2ek−1vk−1ekvk (k ≥ 2) in ~GP(ϕ) is called an
undi-rected semipath (or just semipath) if v0e1v1. . . vk−3ek−2vk−2 is an undirected path in ~GP(ϕ),
ek−1 and ek are distinct directed or undirected edges joining vertices vk−2 and vk−1, and vk−1
and vk, respectively, and ek−1 ∈ {e/ 1, . . . , ek−2}.
Note that since ~GP(ϕ) is a union of disjoint even cycles and 2-cycles, a semipath in ~GP(ϕ)
which does not lie in a 2-cycle is uniquely determined by a sequence of vertices, and we will thus often refer to semipaths as sequences of vertices when there is no risk of confusion.
A semipath Q in ~GP(ϕ) is alternating if for any two consecutive distinct arcs a and a′ in
Q, the head of a is adjacent to the head of a′
, or the tail of a is adjacent to the tail of a′
. A semipath in ~GP(ϕ) is proper if it contains at least two distinct arcs, begins by a red arc and
ends by a red arc, and contains no other red arcs.
Lemma 3.5. Let G be a (3, 5)-biregular graph with an MP -subgraph (M, P ). Suppose further that ϕ is a proper edge coloring of P using colors 1 and 2. There is a correct vertex coloring of ~GP(ϕ) if and only if there is no proper alternating semipath in ~GP(ϕ).
Proof. For necessity, let ψ be a correct vertex coloring of ~GP(ϕ), and suppose, for a
contra-diction, that there is a proper alternating semipath Q = v1v2. . . v2k in ~GP(ϕ). Suppose, for
example, that (v1, v2) is a red arc of ~GP(ϕ) and consequently that ψ(v1) = 5 and ψ(v2) = 3.
(The case when (v2, v1) is an arc of ~GP(ϕ) is similar.) Since v2v3 is a green edge, we have
that ψ(v3) = 5.
If Q contains no blue arcs, then k = 2, (v4, v3) is red and ψ(v3) = 5, which contradicts that
ψ is correct. Suppose now that Q contains at least one blue arc and thus that (v4, v3) is blue.
Then, since ψ(v3) = 5 and ψ is correct, we must have ψ(v4) = 5. In fact, by induction, it is
easy to show that ψ(v2l−1) = ψ(v2l), for each l = 2, . . . , k − 1. If k is even, then (v2k−3, v2k−2)
is a blue arc with ψ(v2k−3) = ψ(v2k−2) = 3, and (v2k, v2k−1) is a red arc. Moreover, since
ψ(v2k−2) = 3, ψ is correct and v2k−2v2k−1 is green, we must have ψ(v2k−1) = 5. However,
since v2k−1 is the head of a red arc, this contradicts that ψ is correct.
If k is odd, then (v2k−2, v2k−3) is a blue arc with ψ(v2k−3) = ψ(v2k−2) = 5, and (v2k−1, v2k)
is a red arc. Moreover, since ψ(v2k−2) = 5, we must have ψ(v2k−1) = 3. Now, since v2k−1 is
Let us now prove sufficiency. Suppose that there is no proper alternating semipath in ~
GP(ϕ). We first prove the following claim.
Claim. There is a vertex coloring ψ of ~GP(ϕ) with colors 3 and 5 satisfying conditions (a)
and (b).
Proof. It suffices to prove that there is such a coloring ψ of a component of ~GP(ϕ). So let T
be a component of ~GP(ϕ). If T has at most one red arc (u, v), then since T is an even cycle,
there is a proper coloring ψ of T with colors 3 and 5 satisfying (a) and (b), and such that ψ(u) = 5 and ψ(v) = 3.
Suppose now that T has at least two red arcs. If T has precisely two red arcs a and a′
, let Q1 and Q2 be two proper semipaths of T , such that a is the first edge of Q1, a′ is the first
edge of Q2, and a and a′ are the only common arcs or edges of Q1 and Q2. Otherwise, if T
contains p ≥ 3 red arcs, let Q1, . . . , Qp be proper semipaths of T , such that Qi and Qi+1 has
precisely one red arc in common, for i = 1, . . . , p, where indices are taken modulo p. We will construct the required coloring ψ by below giving an algorithm for sequentially coloring the vertices of T , when T has p ≥ 3 red arcs. The case when T has 2 red arcs can be dealt with similarly.
First we color the vertices of Q1. Suppose that Q1 = v1(1). . . v (1) k1 and that (v (1) 1 , v (1) 2 )
is the first red arc of Q1. (The case when (v2(1), v (1)
1 ) is the first arc of Q1 is analogous.)
If (vk(1)1−1, v(1)k1 ) is the last arc of Q1, then we color the vertices properly using 3 and 5 and
starting with color 5 at v1(1). Otherwise, (vk(1)1 , vk(1)1−1) is the last arc of Q1, and since Q1 is not
alternating, k1 > 4. We color v1(1), v (1) 3 and v
(1)
4 with color 5 and use color 3 on v (1)
2 . Then
we color the vertices v5(1), . . . , v (1)
k1 properly using colors 3 and 5 and starting with color 3 at
v5(1). Then vk(1)1 is colored 5 and vk(1)1−1 is colored 3, and every green edge on Q1 has one end
colored 3 and one end colored 5, as required.
Suppose now that we have colored all vertices of Q1, . . . , Qj−1 with colors 3 and 5, where
1 < j ≤ p − 1, so that (a) and (b) hold for the hitherto constructed vertex coloring. We now color the vertices of Qj = v
(j) 1 . . . v
(j)
kj. The first two vertices v
(j) 1 , v
(j)
2 are in Qj−1 and thus are
already colored. If the first and last arc on Qj are oriented towards v (j) 2 and v
(j)
kj, respectively,
or towards v1(j) and v(j)k
j−1, respectively, then we color the vertices v
(j) 3 , . . . , v
(j)
kj properly, and
starting with color 3 (5) at v3(j)if v (j)
2 is colored 5 (3); otherwise Qj has length at least 5, and
we color the vertices v3(j) and v4(j) with the same color (i.e. the color in {3, 5} not used on v2(j)), and thereafter color the vertices v5(j), . . . , vk(j)
j properly, and starting with color 3 (5) at
v5(j) if v4(j) is colored 5 (3). This yields a coloring that satisfies (a) and (b) on Q1∪ · · · ∪ Qj.
Finally let us color the uncolored vertices of Qp = v (p) 1 . . . v
(p)
kp. If Qp contains no blue arc,
then Qp has 4 vertices, all of which are already colored. Otherwise, if Qp contains at least
one blue arc, then Qp has length at least 5, and we color vk(p)p−2 with the color in {3, 5} not
used at vk(p)p−1 and similarly, we color v (p)
3 with the color in {3, 5} not used at v (p)
2 . If there
thereafter are any uncolored vertices of Qp, then we color this set of vertices properly using
colors 3 and 5, so that every green edge has one end colored 3 and one end colored 5. This yields a coloring that satisfies (a) and (b) on Qp, and thus on a component T of ~GP(ϕ).
We continue the proof of the lemma. Let ψ be a vertex coloring of ~GP(ϕ) with colors 3
and 5 satisfying conditions (a) and (b), and with the minimum number of blue arcs (x, x′
) such that ψ(x) = 3 and ψ(x′
) = 5; we call such a blue arc ψ-bad.
Suppose that there is some blue arc (u1, v1) satisfying ψ(u1) = 3 and ψ(v1) = 5. We
will prove that there is a coloring ψ′
satisfying (a) and (b) and such that the number of ψ′
-bad arcs is fewer than the number of ψ-bad arcs. Since we assumed that ψ is minimal with respect to this property, this will imply sufficiency of the lemma.
First we prove that there is a path T = u1v1v2. . . v2r−1or T = v1u1u2. . . u2s−1 with origin
at u1 or v1, containing (u1, v1), satisfying that every arc on T is blue except possibly the last
one, and such that either
(i) the two last arcs on T are oriented in such a way that the head of the second to last arc is adjacent to the tail of the last arc, or the tail of the second to last arc is adjacent to the head of the last arc; or
(ii) the last arc (a, b) of T is blue and satisfies ψ(a) 6= ψ(b). We call such a path T good.
Let C be the component containing (u1, v1). Suppose first that C contains no red arc.
Assume further that there is no good path T satisfying (ii) in C. Then since (a) holds and (u1, v1) is ψ-bad, there is an odd number of blue arcs in C. This means that not every pair
of consecutive blue arcs on C are oriented in opposite directions with respect to a cyclic orientation of all edges in the underlying graph of C; that is, there is a good path T in C satisfying (i).
Suppose now that C contains some red arc. Let Q be a proper semipath containing (u1, v1). If Q contains no good path T , then clearly Q is alternating, contradicting that
~
GP(ϕ) contains no proper alternating semipath. We thus conclude that there is a good path
in C. Let T be a minimal good path.
We will assume that T = u1v1v2. . . v2r−1. The case when T = v1u1u2. . . u2s−1 can be
dealt with in an analogous way. Suppose first that (i) holds. Since, T is minimal, we have that ψ(v2i) = ψ(v2i+1), for each i = 1, . . . , r − 2. Moreover, if r is even, then ψ(v2r−2) = 3
and the last arc of T is oriented towards v2r−1; if r is odd, then ψ(v2r−2) = 5 and the last
arc of T is oriented towards v2r−2. Since each red arc of ~GP(ϕ) satisfies condition (b), this
means that the last arc of T must be blue. We define a new coloring ψ′
of ~GP(ϕ) from ψ by switching colors on every green edge
of T and retaining the color of every other vertex of ~GP(ϕ) (so if xy is a green edge of T ,
then we color x with the color in {3, 5} \ ψ(x) and similarly for y). Note that (u1, v1) is not
ψ′
-bad. If the tail of the last arc on T is adjacent to the head of the second to last arc on T , then ψ′
(v2r−2) = 5, and if the head of the last arc on T is adjacent to the tail of the second
to last arc on T , then ψ′
(v2r−2) = 3. This implies that the last arc on T cannot be ψ′-bad.
Moreover, since ψ(v2i) = ψ(v2i+1), ψ′(v2i) = ψ′(v2i+1) for each i = 1, . . . , r − 2. Hence, no arc
on T is ψ′
-bad, which contradicts our assumption about ψ. Hence, there is a correct coloring of ~GP(ϕ).
Suppose now that (ii) holds. Again, we have ψ(v2i) = ψ(v2i+1) for each i = 1, . . . , r − 2. If
to last arc on T is adjacent to the tail of the last arc on T , or that the head of the second to last arc on T is adjacent to the head of the last arc on T . Again, we define the coloring ψ′
from ψ by switching colors on every green edge on T . Trivially, no arc on T is ψ′
-bad, so we conclude that there is a correct coloring of ~GP(ϕ).
The preceding two lemmas imply the following theorem.
Theorem 3.6. Let G be a (3, 5)-biregular graph with an MP -subgraph (M, P ). Then G has an interval 7-coloring if there is a proper edge coloring ϕ with colors 1, 2 of P such that there is no proper alternating semipath in ~GP(ϕ).
Using Theorem 3.6, we can prove that (3, 5)-biregular graphs having a certain kind of MP -subgraph admit an interval coloring.
Corollary 3.7. Let G be a (3, 5)-biregular graph with an MP -subgraph (M, P ). If each component of GP has an even number of maximal (blue, green)-colored paths with an odd
number of blue edges, then G has an interval 7-coloring.
Proof. We will prove that there is a proper edge coloring of ϕ with colors 1 and 2 of P such
that ~GP(ϕ) has no proper alternating semipath. Theorem 3.6 then implies the result.
Any proper edge coloring with colors 1 and 2 of P , induces an orientation of all red and blue edges in GP yielding the graph ~GP(ϕ). As noted above, since ~GP(ϕ) is uniquely defined
by GP and a proper coloring ϕ of P with colors 1 and 2, any orientation D of the red edges
in GP induces a proper edge coloring of P with colors 1 and 2. Such an orientation D of
the red edges thus also induces an orientation of the blue edges in GP. Hence, it suffices
to prove that there is an orientation D of the red edges in GP, such that if ϕ is the edge
coloring of P induced by D, then ~GP(ϕ) has no proper alternating semipath. We call this a
good orientation of the red edges.
A good orientation of the red edges in GP can be obtained as follows. Consider a
compo-nent C in GP and suppose that C has at least one red edge ab. Let B be a cyclic orientation
of all edges in C. First we orient all red edges of C cyclically according to B. For each red arc (c, d) such that there is an odd number of blue edges between c and a, when edges are traversed according to B, we change the direction of (c, d). Denote the obtained mixed graph by H.
Note that since C has an even number of maximal (blue, green)-colored paths with an odd number of blue edges, if uv and xy are consecutive red edges in C (with respect to B), then the arcs corresponding to uv and xy in H are both oriented clockwise or counter-clockwise (with respect to B), if there is an even number of blue edges between uv and xy; and exactly one of these arcs is oriented clockwise (with respect to B), if there is an odd number of blue edges between uv and xy (when edges are traversed in any of the two possible directions).
By repeating the above procedure for every component of GP, we obtain a mixed graph
D with red arcs. Let ϕ be the proper edge coloring of P induced by D, and suppose, for a contradiction, that ~GP(ϕ) contains a proper alternating semipath T = v1. . . vk. Let B
be a cyclic orientation of the edges in the component of GP containing v1, . . . , vk. If T has
an even number of blue arcs, then the first and last arc of T are both oriented clockwise or counter-clockwise (with respect to B), which contradicts that T is alternating. On the other hand, if T contains an odd number of blue arcs, then T contains two distinct red arcs
and exactly one of them is oriented clockwise with respect to B. This contradicts that T is alternating. Hence, there is no proper alternating semipath in ~GP(ϕ), and G therefore has
an interval 7-coloring.
The following is our main result on existence of interval colorings of (3, 5)-biregular graphs. Theorem 3.8. If a (3, 5)-biregular graph has a proper MP -subgraph, then it has an interval
7-coloring.
Proof. Let (M, P ) be a proper MP -subgraph of G. Similarly to the proof of the preceding
corollary it suffices to show that there is an orientation D of the red edges in GP such that
if ϕ is the proper coloring of P induced by D, then there is no proper alternating semipath in ~GP(ϕ).
Since (M, P ) is a proper MP -subgraph of G, we have that for any component Q of G − M ∪ E(P ) which is a path, either both endpoints of Q are endpoints of components in P or both endpoints of Q are internal vertices of components in P . This implies that no component of GP contains both blue and red edges. We construct the required orientation D
of the red edges in GP, for any component C of GP with at least one red edge, by orienting
the red edges cyclically along C. Since any component of GP with at least one red edge has
no blue edges, this orientation induces a coloring ϕ of P , such that ~GP(ϕ) has no proper
alternating semipath. Theorem 3.6 now implies that G has an interval 7-coloring.
4
Conditions for proper
M P -subgraphs in (3, 5)-biregular
graphs
The smallest (3, 5)-biregular graph is the complete bipartite graph K3,5; it has a proper
MP -subgraph.
Lemma 4.1. K3,5 has a proper MP -subgraph.
Proof. Let X = {x1, x2, x3, x4, x5} and Y = {y1, y2, y3} be the partite sets of K3,5. Let ϕ be
the interval 7-coloring of K3,5 defined by setting ϕ(xiyj) = i + j − 1 for all edges xiyj. Let
M be the set of edges colored 4, and let P be the subgraph induced by edges colored 1, 2, 6 or 7. Then (M, P ) is a proper MP -subgraph.
Using the next lemma, we can generate infinitely many connected (3, 5)-biregular graphs having proper MP -subgraphs.
Lemma 4.2. For i ∈ {1, 2}, let Gi be a 2-edge-connected (3, 5)-biregular graph having a
proper MP -subgraph (Mi, Pi), and choose ei ∈ Mi. Let G be the (3, 5)-biregular graph
ob-tained from the disjoint union of G1 and G2 by deleting e1 and e2 and replacing them with two
other edges e′ 1 and e
′
2 joining their appropriate endpoints. Then G is a larger 2-edge-connected
(3, 5)-biregular graph having a proper MP -subgraph.
Proof. It is easily checked that G is 2-edge-connected. Set P = P1 ∪ P2 and
M = M1∪ M2∪ {e′1, e ′
2} \ {e1, e2}.
Next, we give a sufficient condition for a (3, 5)biregular graph to have a proper MP -subgraph. For a matching M in such a graph G = (X, Y ; E), denote by XM the vertices in
X covered by M.
Theorem 4.3. Let G = (X, Y ; E) be a (3, 5)-biregular graph. If there is an Y -saturating matching M with the property that there is a partition Y = Y1∪ Y2 such that
(i) G[X \ XM ∪ Y1] − M is (2, 4)-biregular, and
(ii) G[XM ∪ Y2] − M is (2, 3)-biregular,
then G has a proper MP -subgraph (and thus an interval 7-coloring).
Proof. Let G = (X, Y ; E) be a (3, 5)-biregular graph and M a Y -saturating matching. Set
X′
= X \ XM and H = G − M. Suppose that there is a partition Y = Y1∪ Y2 such that
H[X′
∪Y1] is (2, 4)-biregular and H[XM∪Y2] is (2, 3)-biregular. This implies that H[XM∪Y1]
is empty and H[X′∪ Y
2] is 1-regular. We shall construct a subgraph P of H such that (M, P )
is a proper MP -subgraph of G. Since H[X′
∪ Y1] is (2, 4)-biregular it has a (1, 2)-biregular subgraph J (e.g. by taking
every second edge in an eulerian circuit in H[X′∪ Y
1]). Let M1 be a matching in H[XM∪ Y2]
covering all vertices of Y2. Such a matching exists by Hall’s condition. Let XM1 be the
vertices of X covered by M1. Now we construct P by setting
V (P ) = X′
∪ Y1∪ Y2∪ XM1 and E(P ) = E(J) ∪ E(H[X′∪ Y2]) ∪ M1.
Consider a component F in P . Exactly one vertex y of F lies in Y1 and y is adjacent to
two vertices x1, x2 ∈ X′ in F . Each of the vertices x1, x2 is adjacent to exactly one vertex
y′ ∈ Y
2 in F , y′ is adjacent to exactly one vertex xM ∈ XM1 ⊆ XM in F , and xM has degree
one in P . Hence, each component in P is a path of length 6 with endpoints in X. Consider a vertex x ∈ X. If x is not covered by M, then x ∈ X′
and each vertex in X′
has degree 2 in P . We conclude that (M, P ) is an MP -subgraph of G.
Suppose now that Q is a component in G − M ∪ E(P ) which is a path and one endpoint of Q is in XM. Since X′ and XM lie in different components of G − M ∪ E(P ), and each
vertex of Y2 has degree 2 in G − M ∪ E(P ), both endpoints of Q are in XM. Hence, (M, P )
is proper, and by Theorem 3.8, G has an interval coloring.
A (3, 5)-biregular graph satisfying the hypothesis of Theorem 4.3 is the disjoint union of a (2, 4)-biregular graph H1, a (2, 3)-biregular H2 and two matchings. Since, for any integer
b, a (2, b)-biregular graph can be obtained from a b-regular graph H by subdividing each edge in H, Theorem 4.3 yields an infinite family of (3, 5)-biregular graphs having proper MP -subgraphs. On the other hand, it is not hard to construct examples of (3, 5)-biregular graphs that do not have MP -subgraphs. Consider, for example, the graph G = (X, Y ; E) in Figure 1.
Suppose that there is a subgraph F of G that covers all vertices of degree 5 and such that all components of F are paths of length 6 with endpoints at 3-valent vertices. Denote the components of G − u by G1, G2 and G3, respectively. A path through u contains vertices
u
Figure 1: A (3, 5)-biregular graph having no MP -subgraph.
Y ∩ V (G1) are in components of F that do not pass through u. But since |Y ∩ V (G1)| = 5,
this is impossible. Hence, there is no subgraph F of G that covers all vertices of degree 5 and such that all components of F are paths of length 6 with endpoints at 3-valent vertices, which clearly implies that G has no MP -subgraph.
Finally, we note the following condition for interval colorings of (3, 5)-biregular graphs. The proof is a simple application of the technique in [25].
Proposition 4.4. Let G = (X, Y ; E) be a (3, 5)-biregular graph. If there are two disjoint sets X′ ⊆ X and X′′ ⊆ X, such that
(i) NG(X′) = NG(X′′) = Y ;
(ii) if x, x′ ∈ X′ or x, x′ ∈ X′′, then N
G(x) ∩ NG(x′) = ∅;
(iii) if x ∈ X′′
, then there is a vertex x′
∈ X′
, such that NG(x) = NG(x′);
then G has an interval 7-coloring.
Proof. Let G = (X, Y ; E) be a (3, 5)-biregular graph and suppose that there are disjoint sets
X′ ⊆ X and X′′ ⊆ X, satisfying (i)-(iii). Then G − X′ − X′′
is 3-regular and G − X′′
is (3, 4)-biregular. Pyatkin [25] proved that there is an interval 6-coloring ϕ of G − X′′ such
that ϕ(x) = {4, 5, 6} for each vertex x ∈ X′
and if NG(x) = {y1, y2, y3}, where ϕ(xyi) = i + 3,
then ϕ(yi) = {i, . . . , i + 3}, i = 1, 2, 3.
Note that each edge e′′ of G[X′′∪ Y ] is adjacent to exactly one edge e′ of G[X′∪ Y ]. We
define an edge coloring ψ of G[X′′
∪ Y ] by setting ψ(e′′
) = ϕ(e′
) + 1 for each such edge e′′
of G[X′′∪ Y ]. The colorings ψ and ϕ together constitute an interval 7-coloring of G.
References
[1] A.S. Asratian, T.M.J. Denley, R. H¨aggkvist, Bipartite Graphs and their Applications, Cambridge University Press, Cambridge, 1998.
[2] A.S. Asratian, C.J. Casselgren, On interval edge colorings of (α, β)-biregular bipartite graphs, Discrete Math. 307 (2007) 1951–1956.
[3] A.S. Asratian, C.J. Casselgren, On path factors of (3, 4)-biregular bigraphs, Graphs Combin. 24 (2008) 405–411.
[4] A.S. Asratian, C.J. Casselgren, J. Vandenbussche, D.B. West, Proper path factors and interval edge-coloring of (3, 4)-biregular bigraphs, J. Graph Theory 61 (2009) 88–97. [5] A.S. Asratian, R.R. Kamalian, Investigation of interval edge-colorings of graphs, JCT
B 62 (1994) 34–43.
[6] A.S. Asratian, R.R. Kamalian, Interval coloring of the edges of a multigraph (in Russian), Applied Mathematics 5, Yerevan University, (1987), 25–34.
[7] M.A. Axenovich, On interval colorings of planar graphs, Congr. Numer. 159 (2002) 77–94.
[8] C.J. Casselgren, A note on path factors of (3, 4)-biregular bigraphs, Electron. J. Comb. 18 (2011) #P218.
[9] C.J. Casselgren, B. Toft, On interval edge colorings of biregular bipartite graphs with
small vertex degrees, J. Graph Theory 80 (2015) 83–97.
[10] C.J. Casselgren, Some results on interval edge colorings of bipartite graphs, Master’s thesis, Link¨oping University, 2005.
[11] K. Giaro, The complexity of consecutive ∆-coloring of bipartite graphs: 4 is easy, 5 is hard, Ars Combin. 47 (1997) 287–298.
[12] K. Giaro, M. Kubale, Compact scheduling of zero-one time operations in multi-stage systems, Disc. Appl. Math. 145 (2004) 95–103
[13] K. Giaro, M. Kubale, Consecutive edge-colorings of complete and incomplete Cartesian products of graphs, Congr. Numer. 128 (1997) 143–149.
[14] H. M. Hansen, Scheduling with minimum waiting periods (in Danish), Master’s thesis, Odense University, Odense, Denmark, 1992.
[15] D. Hanson, C.O.M. Loten, A lower bound for interval colouring bi-regular bipartite graphs, Bull. ICA 18 (1996) 69–74.
[16] D. Hanson, C.O.M. Loten, B. Toft, On interval colourings of bi-regular bipartite graphs, Ars Combin. 50 (1998) 23–32.
[17] T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley Interscience, 1995.
[18] R. R. Kamalian, Interval colorings of complete bipartite graphs and trees, Preprint, Comp. Cen. of Acad. Sci. of Armenian SSR, Erevan, 1989 (in Russian).
[19] A.V. Kostochka, Unpublished manuscript, 1995.
[20] M. Kubale, A. Nadolski, Chromatic scheduling in a cyclic open shop, Eur. J. Oper. Res. 164 (2005) 585–591.
[21] A. Nadolski, Compact cyclic edge-colorings of graphs, Discrete Math. 308 (2008) 2407– 2417.
[22] O. Ore, Studies in directed graphs I, Ann. Math. 63 (1956) 383–406.
[23] P.A. Petrosyan, H.H. Khachatrian, Interval Non-edge-Colorable Bipartite Graphs and Multigraphs, J. Graph Theory 76 (2014) 200–216.
[24] P.A. Petrosyan, S.T. Mkhitaryan, Interval cyclic edge-colorings of graphs, Discrete Math. 339 (2016) 1848-1860.
[25] A.V. Pyatkin, Interval coloring of (3, 4)-biregular bipartite graphs having large cubic subgraphs, J. Graph Theory 47 (2004) 122–128.
[26] S.V. Sevastjanov, Interval colorability of the edges of a bipartite graph (in Russian), Metody Diskret. Anal., 50 (1990), 61–72.
[27] M. Stiebitz, B. Toft, D. Scheide, L.M. Favrholdt, Graph Edge Colouring: Vizing’s
The-orem and Goldberg’s Conjecture, Wiley Interscience, 2012.
[28] Fan Yang, Xiangwen Li, Interval coloring of (3, 4)-biregular bigraphs having two (2, 3)-biregular bipartite subgraphs, Appl. Math. Lett. 24 (2011) 1574-1577.
